Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 10.3s

Alternatives: 16

Speedup: 2.4×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \frac{\frac{\alpha + 1}{t\_0}}{3 + \left(\alpha + \beta\right)} \cdot \frac{\beta + 1}{t\_0} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (+ alpha beta))))
   (* (/ (/ (+ alpha 1.0) t_0) (+ 3.0 (+ alpha beta))) (/ (+ beta 1.0) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = -2.0 - (alpha + beta);
	return (((alpha + 1.0) / t_0) / (3.0 + (alpha + beta))) * ((beta + 1.0) / t_0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (-2.0d0) - (alpha + beta)
    code = (((alpha + 1.0d0) / t_0) / (3.0d0 + (alpha + beta))) * ((beta + 1.0d0) / t_0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = -2.0 - (alpha + beta);
	return (((alpha + 1.0) / t_0) / (3.0 + (alpha + beta))) * ((beta + 1.0) / t_0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = -2.0 - (alpha + beta)
	return (((alpha + 1.0) / t_0) / (3.0 + (alpha + beta))) * ((beta + 1.0) / t_0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(-2.0 - Float64(alpha + beta))
	return Float64(Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(3.0 + Float64(alpha + beta))) * Float64(Float64(beta + 1.0) / t_0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = -2.0 - (alpha + beta);
	tmp = (((alpha + 1.0) / t_0) / (3.0 + (alpha + beta))) * ((beta + 1.0) / t_0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

   6. lower-+.f64 93.7

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   9. lower-+.f64 93.7

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

   11. metadata-eval 93.7

       \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

4. Applied rewrites 93.7%

   \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

6. Applied rewrites 92.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   4. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\color{blue}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\color{blue}{\left(2 + 1\right)} + \left(\beta + \alpha\right)\right)} \]

   8. associate-+r+ N/A

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(2 + \left(1 + \left(\beta + \alpha\right)\right)\right)}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(2 + \color{blue}{\left(1 + \left(\beta + \alpha\right)\right)}\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right)}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right)}} \]

   12. times-frac N/A

       \[\leadsto \color{blue}{\frac{1 + \beta}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1 + \beta}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 + \beta}{-2 - \left(\beta + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{1 + \beta}{-2 - \left(\beta + \alpha\right)} \cdot \color{blue}{\frac{\frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1 + \beta}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{-2 - \left(\beta + \alpha\right)}}{3 + \left(\beta + \alpha\right)}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{\frac{\alpha + 1}{-2 - \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)} \cdot \frac{\beta + 1}{-2 - \left(\alpha + \beta\right)} \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 10^{+79}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{t\_0}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (+ alpha beta))))
   (if (<= beta 1e+79)
     (/ (* (+ alpha 1.0) (/ (+ beta 1.0) t_0)) (* (+ 3.0 (+ alpha beta)) t_0))
     (/ (/ (+ alpha 1.0) (+ (+ 3.0 alpha) beta)) (+ (+ 2.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = -2.0 - (alpha + beta);
	double tmp;
	if (beta <= 1e+79) {
		tmp = ((alpha + 1.0) * ((beta + 1.0) / t_0)) / ((3.0 + (alpha + beta)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) - (alpha + beta)
    if (beta <= 1d+79) then
        tmp = ((alpha + 1.0d0) * ((beta + 1.0d0) / t_0)) / ((3.0d0 + (alpha + beta)) * t_0)
    else
        tmp = ((alpha + 1.0d0) / ((3.0d0 + alpha) + beta)) / ((2.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = -2.0 - (alpha + beta);
	double tmp;
	if (beta <= 1e+79) {
		tmp = ((alpha + 1.0) * ((beta + 1.0) / t_0)) / ((3.0 + (alpha + beta)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = -2.0 - (alpha + beta)
	tmp = 0
	if beta <= 1e+79:
		tmp = ((alpha + 1.0) * ((beta + 1.0) / t_0)) / ((3.0 + (alpha + beta)) * t_0)
	else:
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(-2.0 - Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1e+79)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(Float64(beta + 1.0) / t_0)) / Float64(Float64(3.0 + Float64(alpha + beta)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + alpha) + beta)) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = -2.0 - (alpha + beta);
	tmp = 0.0;
	if (beta <= 1e+79)
		tmp = ((alpha + 1.0) * ((beta + 1.0) / t_0)) / ((3.0 + (alpha + beta)) * t_0);
	else
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+79], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999967e78

   1. Initial program 98.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 98.8

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 98.8

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 98.8

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}}{-2 - \left(\beta + \alpha\right)}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{1 + \beta}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{1 + \beta}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

      6. lower-/.f64 99.8

         \[\leadsto \frac{\left(1 + \alpha\right) \cdot \color{blue}{\frac{1 + \beta}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{1 + \beta}{-2 - \left(\beta + \alpha\right)}}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)} \]

   if 9.99999999999999967e78 < beta

   1. Initial program 79.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 89.9

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 89.9%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   7. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+79}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{-2 - \left(\alpha + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \beta\right) + \alpha\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(t\_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 beta) alpha)))
   (if (<= beta 2e+20)
     (/ (* (+ alpha 1.0) (+ beta 1.0)) (* (* t_0 (+ 3.0 (+ alpha beta))) t_0))
     (/ (/ (+ alpha 1.0) (+ (+ 3.0 alpha) beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double tmp;
	if (beta <= 2e+20) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 * (3.0 + (alpha + beta))) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + beta) + alpha
    if (beta <= 2d+20) then
        tmp = ((alpha + 1.0d0) * (beta + 1.0d0)) / ((t_0 * (3.0d0 + (alpha + beta))) * t_0)
    else
        tmp = ((alpha + 1.0d0) / ((3.0d0 + alpha) + beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double tmp;
	if (beta <= 2e+20) {
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 * (3.0 + (alpha + beta))) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (2.0 + beta) + alpha
	tmp = 0
	if beta <= 2e+20:
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 * (3.0 + (alpha + beta))) * t_0)
	else:
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + beta) + alpha)
	tmp = 0.0
	if (beta <= 2e+20)
		tmp = Float64(Float64(Float64(alpha + 1.0) * Float64(beta + 1.0)) / Float64(Float64(t_0 * Float64(3.0 + Float64(alpha + beta))) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + alpha) + beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (2.0 + beta) + alpha;
	tmp = 0.0;
	if (beta <= 2e+20)
		tmp = ((alpha + 1.0) * (beta + 1.0)) / ((t_0 * (3.0 + (alpha + beta))) * t_0);
	else
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]}, If[LessEqual[beta, 2e+20], N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e20

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 99.9

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 99.9

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   6. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   if 2e20 < beta

   1. Initial program 81.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 85.6

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 85.6%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   7. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{2 + \beta}}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8e+15)
   (/ (/ (+ beta 1.0) (+ 2.0 beta)) (* (+ 3.0 beta) (+ 2.0 beta)))
   (/ (/ (+ alpha 1.0) (+ (+ 3.0 alpha) beta)) (+ (+ 2.0 beta) alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+15) {
		tmp = ((beta + 1.0) / (2.0 + beta)) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d+15) then
        tmp = ((beta + 1.0d0) / (2.0d0 + beta)) / ((3.0d0 + beta) * (2.0d0 + beta))
    else
        tmp = ((alpha + 1.0d0) / ((3.0d0 + alpha) + beta)) / ((2.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+15) {
		tmp = ((beta + 1.0) / (2.0 + beta)) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8e+15:
		tmp = ((beta + 1.0) / (2.0 + beta)) / ((3.0 + beta) * (2.0 + beta))
	else:
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(2.0 + beta)) / Float64(Float64(3.0 + beta) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + alpha) + beta)) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8e+15)
		tmp = ((beta + 1.0) / (2.0 + beta)) / ((3.0 + beta) * (2.0 + beta));
	else
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / ((2.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.8e15

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 15.1

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 15.1%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 32.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 82.7

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 82.7%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(3 + \beta\right)} \cdot \left(2 + \beta\right)} \]

       4. lower-+.f64 62.4

          \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\left(3 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

   13. Applied rewrites 62.4%

       \[\leadsto \frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

   if 4.8e15 < beta

   1. Initial program 81.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 84.7

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 84.7%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   7. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{2 + \beta}}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \beta\right) + \alpha\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\left(3 + \beta\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 beta) alpha)))
   (if (<= beta 5e+16)
     (/ (+ beta 1.0) (* (* (+ 3.0 beta) t_0) t_0))
     (/ (/ (+ alpha 1.0) (+ (+ 3.0 alpha) beta)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta + 1.0) / (((3.0 + beta) * t_0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + beta) + alpha
    if (beta <= 5d+16) then
        tmp = (beta + 1.0d0) / (((3.0d0 + beta) * t_0) * t_0)
    else
        tmp = ((alpha + 1.0d0) / ((3.0d0 + alpha) + beta)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta + 1.0) / (((3.0 + beta) * t_0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (2.0 + beta) + alpha
	tmp = 0
	if beta <= 5e+16:
		tmp = (beta + 1.0) / (((3.0 + beta) * t_0) * t_0)
	else:
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + beta) + alpha)
	tmp = 0.0
	if (beta <= 5e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(Float64(3.0 + beta) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + alpha) + beta)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (2.0 + beta) + alpha;
	tmp = 0.0;
	if (beta <= 5e+16)
		tmp = (beta + 1.0) / (((3.0 + beta) * t_0) * t_0);
	else
		tmp = ((alpha + 1.0) / ((3.0 + alpha) + beta)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]}, If[LessEqual[beta, 5e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(N[(3.0 + beta), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 63.4

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]

   5. Applied rewrites 63.4%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{3 + \beta}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{3 + \beta} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(3 + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(3 + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(3 + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(3 + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \alpha, \beta, 1 + \alpha\right)}{\left(\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 82.5

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 82.5%

       \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 5e16 < beta

   1. Initial program 81.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 84.7

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 84.7%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   7. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \beta\right) + \alpha\\ t_1 := \left(3 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 24:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 beta) alpha)) (t_1 (+ (+ 3.0 alpha) beta)))
   (if (<= beta 24.0)
     (/ (fma 0.25 beta 0.5) (* t_0 t_1))
     (/ (/ (+ alpha 1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double t_1 = (3.0 + alpha) + beta;
	double tmp;
	if (beta <= 24.0) {
		tmp = fma(0.25, beta, 0.5) / (t_0 * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + beta) + alpha)
	t_1 = Float64(Float64(3.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 24.0)
		tmp = Float64(fma(0.25, beta, 0.5) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 24.0], N[(N[(0.25 * beta + 0.5), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 24

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 14.2

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 14.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 83.0

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 83.0%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right)}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 24 < beta

   1. Initial program 83.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 81.1

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 81.1%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   7. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 24:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(3 + \alpha\right) + \beta}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 40.0)
   (/ (fma 0.25 beta 0.5) (* (+ (+ 2.0 beta) alpha) (+ (+ 3.0 alpha) beta)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ (+ alpha beta) 1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 40.0) {
		tmp = fma(0.25, beta, 0.5) / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 40.0)
		tmp = Float64(fma(0.25, beta, 0.5) / Float64(Float64(Float64(2.0 + beta) + alpha) * Float64(Float64(3.0 + alpha) + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 40.0], N[(N[(0.25 * beta + 0.5), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 40

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 14.2

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 14.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 83.0

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 83.0%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right)}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 40 < beta

   1. Initial program 83.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1 + \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot 1}} \]

      6. lower-+.f64 83.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\alpha + \beta\right)\right)} + 2 \cdot 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\alpha + \beta\right)}\right) + 2 \cdot 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      9. lower-+.f64 83.0

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \color{blue}{\left(\beta + \alpha\right)}\right) + 2 \cdot 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2 \cdot 1}} \]

      11. metadata-eval 83.0

          \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(1 + \left(\beta + \alpha\right)\right) + \color{blue}{2}} \]

   4. Applied rewrites 83.0%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(1 + \left(\beta + \alpha\right)\right) + 2}} \]

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

      2. lower-+.f64 80.5

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]

   7. Applied rewrites 80.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(1 + \left(\beta + \alpha\right)\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 40.0)
   (/ (fma 0.25 beta 0.5) (* (+ (+ 2.0 beta) alpha) (+ (+ 3.0 alpha) beta)))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 40.0) {
		tmp = fma(0.25, beta, 0.5) / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 40.0)
		tmp = Float64(fma(0.25, beta, 0.5) / Float64(Float64(Float64(2.0 + beta) + alpha) * Float64(Float64(3.0 + alpha) + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 40.0], N[(N[(0.25 * beta + 0.5), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 40

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 14.2

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 14.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 83.0

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 83.0%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right)}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 40 < beta

   1. Initial program 83.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 73.3

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]

   5. Applied rewrites 73.3%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]

   6. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]

      2. lower-+.f64 80.3

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{3 + \beta} \]

   8. Applied rewrites 80.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 48.0)
   (/ 0.5 (* (+ (+ 2.0 beta) alpha) (+ (+ 3.0 alpha) beta)))
   (/ (/ (+ alpha 1.0) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 48.0d0) then
        tmp = 0.5d0 / (((2.0d0 + beta) + alpha) * ((3.0d0 + alpha) + beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 48.0:
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta))
	else:
		tmp = ((alpha + 1.0) / beta) / (3.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 48.0)
		tmp = Float64(0.5 / Float64(Float64(Float64(2.0 + beta) + alpha) * Float64(Float64(3.0 + alpha) + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 48.0)
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	else
		tmp = ((alpha + 1.0) / beta) / (3.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 48.0], N[(0.5 / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 48

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 14.2

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 14.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 83.0

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 83.0%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.7%

       \[\leadsto \frac{0.5}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 48 < beta

   1. Initial program 83.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 73.3

         \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]

   5. Applied rewrites 73.3%

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{3 + \beta}} \]

   6. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]

      2. lower-+.f64 80.3

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{3 + \beta} \]

   8. Applied rewrites 80.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 48.0)
   (/ 0.5 (* (+ (+ 2.0 beta) alpha) (+ (+ 3.0 alpha) beta)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 48.0d0) then
        tmp = 0.5d0 / (((2.0d0 + beta) + alpha) * ((3.0d0 + alpha) + beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 48.0:
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 48.0)
		tmp = Float64(0.5 / Float64(Float64(Float64(2.0 + beta) + alpha) * Float64(Float64(3.0 + alpha) + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 48.0)
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((3.0 + alpha) + beta));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 48.0], N[(0.5 / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 48

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 14.2

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 14.2%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

      3. lower-+.f64 83.0

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   10. Applied rewrites 83.0%

       \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{2}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.7%

       \[\leadsto \frac{0.5}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 48 < beta

   1. Initial program 83.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 77.8

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.3%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(3 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\alpha + 1}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+77)
   (/ (+ alpha 1.0) (* (+ 3.0 beta) (+ 2.0 beta)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+77) {
		tmp = (alpha + 1.0) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+77) then
        tmp = (alpha + 1.0d0) / ((3.0d0 + beta) * (2.0d0 + beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+77) {
		tmp = (alpha + 1.0) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+77:
		tmp = (alpha + 1.0) / ((3.0 + beta) * (2.0 + beta))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+77)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + beta) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+77)
		tmp = (alpha + 1.0) / ((3.0 + beta) * (2.0 + beta));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+77], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000004e77

   1. Initial program 98.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. lower-+.f64 20.6

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Applied rewrites 20.6%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(3 + \beta\right)} \cdot \left(2 + \beta\right)} \]

      4. lower-+.f64 19.2

         \[\leadsto \frac{1 + \alpha}{\left(3 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

   10. Applied rewrites 19.2%

       \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}} \]

   if 5.00000000000000004e77 < beta

   1. Initial program 79.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 86.1

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.6%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\alpha + 1}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.9e+151)
   (/ (+ alpha 1.0) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9e+151) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.9d+151) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9e+151) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.9e+151:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.9e+151)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.9e+151)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.9e+151], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9e151

   1. Initial program 97.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 18.2

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 1.9e151 < beta

   1. Initial program 74.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 88.4

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.4%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.5%

       \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\frac{\alpha + 1}{\beta}}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (/ (+ alpha 1.0) beta) beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) / beta) / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((alpha + 1.0) / beta) / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(alpha + 1.0) / beta) / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / beta) / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 30.8

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 30.8%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
6. Step-by-step derivation

7. Applied rewrites 31.7%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

8. Final simplification 31.7%

   \[\leadsto \frac{\frac{\alpha + 1}{\beta}}{\beta} \]
9. Add Preprocessing

Alternative 14: 52.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5e-11) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-11) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5d-11) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-11) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 5e-11:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5e-11)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5e-11)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 5e-11], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.00000000000000018e-11

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 39.2

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.0%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]

   if 5.00000000000000018e-11 < alpha

   1. Initial program 83.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 16.7

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha + 1}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (+ alpha 1.0) (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (alpha + 1.0) / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (alpha + 1.0d0) / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (alpha + 1.0) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (alpha + 1.0) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(alpha + 1.0) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (alpha + 1.0) / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 30.8

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 30.8%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Final simplification 30.8%

   \[\leadsto \frac{\alpha + 1}{\beta \cdot \beta} \]
7. Add Preprocessing

Alternative 16: 31.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ alpha (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = alpha / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return alpha / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(alpha / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = alpha / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 30.8

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 30.8%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around inf

   \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 19.8%

   \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))